and many more benefits!

Find us on Facebook

GMAT Club Timer Informer

Hi GMATClubber!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Hide Tags

A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

30 Aug 2014, 04:58

1

This post receivedKUDOS

16

This post wasBOOKMARKED

00:00

A

B

C

D

E

Difficulty:

85% (hard)

Question Stats:

52%(01:52) correct 48%(02:12) wrong based on 269 sessions

HideShow timer Statistics

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

08 Sep 2014, 07:42

2

This post receivedKUDOS

PathFinder007 wrote:

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A 20B 45C 60D 55E 77

Assume this as an equilateral triangle with side as a (Yes I know, this isn't an equilateral triangle hence i asked you to assume, this is the only diagram I got from internet and I am too lazy to draw one myself)

so now area of an equilateral triangle = \(\frac{\sqrt3}{4} * a^2\) -----1and area of the triangle is also equal to Area of triangle AOC+Area of AOB + Area of BOC = \(\frac{1}{2} * a * r\)( r is height of individual triangle) ---------2

from equation 1 & 2 above

\(\frac{\sqrt3}{4} * a^2 = 3 * \frac{1}{2} * a * r\)

from here we can get the value of a i.e. \(a = 2\sqrt{3} * r\) ---------3

Now, In the question we need to find out \(\frac{Area of circle inscribed}{Area of the equilateral triangle}\)

which is equal to \(\frac{{\pi * r^2}}{{3* 1/2 * (2 \sqrt3 r)^2}}\) ------substituting the value of a from equation 3

=\(\frac{\pi}{{3 * \sqrt3}}\)

\(\approx \frac{3.14}{{3 * 1.72}}\)

\(\approx \frac{3.14}{{3 * 1.72}}\)

\(\approx \frac{2}{3} \approx 0.66\)

Notice that we reduced the numerator by \(0.26\) so our answer is going to be a bit inflated.Looking at the answer choices, C is the closest. (Notice that there's nothing between 60 and 70 in the options so we can be a little imprecise in this case)

Hence the solution is C
_________________

The buttons on the left are the buttons you are looking for

Last edited by Anamika2014 on 09 Sep 2014, 09:27, edited 1 time in total.

A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

08 Sep 2014, 20:26

9

This post receivedKUDOS

PathFinder007 wrote:

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

Re: A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

08 Sep 2014, 20:46

Anamika2014 wrote:

PathFinder007 wrote:

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A 20B 45C 60D 55E 77

Assume this as an equilateral triangle with side as a (Yes I know, this isn't an equilateral triangle hence i asked you to assume, this is the only diagram I got from internet and I am too lazy to draw one myself)

Attachment:

Su55k02_m10 (1).gif

so now area of an equilateral triangle = \(\frac{\sqrt3}{4} * a^2\) -----1and area of the triangle is also equal to Area of triangle AOC+Area of AOB + Area of BOC = \(\frac{1}{2} * a * r\)( r is height of individual triangle) ---------2

from equation 1 & 2 above

\(\frac{\sqrt3}{4} * a^2 = 3 * \frac{1}{2} * a * r\)

from here we can get the value of a i.e. \(a = 2\sqrt{3} * r\) ---------3

Now, In the question we need to find out \(\frac{Area of circle inscribed}{Area of the equilateral triangle}\)

which is equal to \(\frac{{\pi * r^2}}{{3* 1/2 * (2 \sqrt3 r)^2}}\) ------substituting the value of a from equation 3

=\(\frac{\pi}{{3 * \sqrt3}}\)

\approx \frac{3.14}{{3 * 1.72}}

\(\approx \frac{3.4}{{3 * 1.72}}

[m]\approx \frac{2}{3} \approx 0.66\)

Notice that we reduced the numerator by \(0.26\) so our answer is going to be a bit inflated.Looking at the answer choices, C is the closest. (Notice that there's nothing between 60 and 70 in the options so we can be a little imprecise in this case)

Show Tags

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A 20B 45C 60D 55E 77

PareshGmat's solution is crisp and perfect. Let me just add the explanation for this:

Radius of inscribed circle =\(\frac{\sqrt{3}a}{6}\)

We know that the altitude of the equilateral triangle will be \(\frac{\sqrt{3}a}{2}\)

The altitude will also be the median and will pass through the center of the circle (since it is an equilateral triangle). We know that centroid divides the median in the ratio 2:1. The centroid will be the center of the circle since each median will pass through it due to symmetry. Hence the radius of the circle will be one third of the altitude.

Re: A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

03 Feb 2016, 04:03

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Show Tags

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A 20B 45C 60D 55E 77

To answer this question we need (Area of circle/Area of Triangle)*100

If the circle is in Equilateral triangle then (1/3)*Height of equilateral Triangle = Radius of Circle

Prosper!!!GMATinsightBhoopendra Singh and Dr.Sushma Jhae-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772Online One-on-One Skype based classes and Classroom Coaching in South and West Delhihttp://www.GMATinsight.com/testimonials.html

Re: A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

10 Mar 2017, 02:43

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

10 Mar 2017, 09:07

PathFinder007 wrote:

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A 20B 45C 60D 55E 77

let r=both the radius of the inscribed circle and the base of each of the 6 identical 30-60-90 right triangles subsumed by the equilateral trianglearea of circle=⫪r^2area of equilateral triangle=6*1/2*r*r√3=3r^2√3⫪r^2/3r^2√3=⫪/3√3≈.60=60%C

Re: A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

14 Mar 2017, 22:44

PareshGmat wrote:

PathFinder007 wrote:

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

How did you know to apply these formulas? Was there some kind of algebra you did to arrive at root3 divided by 4? I don't understand where you got root 3 divided 4 - I know it works because I plugged it in and tried but don't understand what math I need to do to get there.

Show Tags

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

How did you know to apply these formulas? Was there some kind of algebra you did to arrive at root3 divided by 4? I don't understand where you got root 3 divided 4 - I know it works because I plugged it in and tried but don't understand what math I need to do to get there.

If the side of an equilateral triangle is s,

The altitude of the equilateral triangle \(= \frac{\sqrt{3}}{2} * s\)(Draw the altitude of an equilateral triangle and then use pythagorean theorem on the right triangle obtained)

Re: A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

15 Mar 2017, 17:39

Okay the formulas appear to be more clear and logical now- but my question is how do you divide a number by an imperfect square- for example, in a equilateral triangle with sides of 2, I used the formula root divided by 4 times 2^2 to get the area. The formula needed to solve the problem as stated is area of circle/ area of triangle * 100- so now we have the denominator. Knowing that 1/3 of the height of equilateral triangle is the radius- we can use the Pythagorean Theorem or other formula Karishma stated and arrive at root/ three. The area of the circle is then pi times root/3 squared. We can then simplify and arrive at pi times 1/3 (1/3 is the radius square in simplest form) divided by root 3. On my calculator this equals .60- though how would you make that calculation without a calculator?

Re: A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

15 Mar 2017, 21:50

1

This post receivedKUDOS

Top Contributor

1

This post wasBOOKMARKED

Hi,Since couple of replies already used equilateral triangle formula, to solve the questionLet’s see now how to solve this question, with using the basic area of triangle formula,If the question asks for percentages or ratios always we can try some number out,Here say the side of the equilateral triangle is “6”,We have to find the area of the triangle and area of the circle inscribed,That is,(Area of the circle inscribed/Area of the triangle)*100Area of the circle is (pi)* r^2Area of the triangle is ½ * base * height

Refer to the diagram,

So, here area of the triangle, ½ * 6 * 3 (root 3) = 9 root 3,Also remember that, radius of the circle is always 1/3 rd the height of the equilateral triangle (This is because, Medians of the triangle intersects at 1:2 ratio).So here the radius would be, 1/3(3 root 3) = root 3So the Area of the circle is, pi * (root 3)^2 = 3 * pi,We can approximate “pi” value as “3“, because answer choices are wide enough, So then area of the circle approximately is 9.Now, Area of the triangle, we can approximate as 9 root 3 = 9 * 1.7 = 15.3, So we can further approximate it as 15.So we can find the percentage now,(Area of the circle inscribed/Area of the triangle)*100(9/15)* 100.This is 60%.So the answer is C.Hope this helps

Re: A circle is inscribed in an equilateral triangle, Find area [#permalink]

Show Tags

16 Mar 2017, 06:01

VeritasPrepKarishma wrote:

PathFinder007 wrote:

A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A 20B 45C 60D 55E 77

PareshGmat's solution is crisp and perfect. Let me just add the explanation for this:

Radius of inscribed circle =\(\frac{\sqrt{3}a}{6}\)

We know that the altitude of the equilateral triangle will be \(\frac{\sqrt{3}a}{2}\)

The altitude will also be the median and will pass through the center of the circle (since it is an equilateral triangle). We know that centroid divides the median in the ratio 2:1. The centroid will be the center of the circle since each median will pass through it due to symmetry. Hence the radius of the circle will be one third of the altitude.

Radius = \(\frac{\sqrt{3}a}{2} * \frac{1}{3} = \frac{\sqrt{3}a}{6}\)

PareshGmat's solution is comprehensive ..Could you please give some similar problems Like it...I think this is harder than OG books questions.

Show Tags

Okay the formulas appear to be more clear and logical now- but my question is how do you divide a number by an imperfect square- for example, in a equilateral triangle with sides of 2, I used the formula root divided by 4 times 2^2 to get the area. The formula needed to solve the problem as stated is area of circle/ area of triangle * 100- so now we have the denominator. Knowing that 1/3 of the height of equilateral triangle is the radius- we can use the Pythagorean Theorem or other formula Karishma stated and arrive at root/ three. The area of the circle is then pi times root/3 squared. We can then simplify and arrive at pi times 1/3 (1/3 is the radius square in simplest form) divided by root 3. On my calculator this equals .60- though how would you make that calculation without a calculator?

You are asked for the closest value, so approximate.sqrt(2) = 1.4sqrt(3) = 1.7

If needed, these values are likely to be given in the question anyway.

Though most likely, options in actual GMAT questions will retain the irrational numbers. So you would probably see options in terms of sqrt(3).
_________________